Bifurcation Analysis and Multiobjective Nonlinear Model Predictive Control of Sustainable Ecosystems

Lakshmi Narayan Sridhar

doi:doi:10.11648/j.ijssam.20240903.11

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Bifurcation Analysis and Multiobjective Nonlinear Model Predictive Control of Sustainable Ecosystems

Lakshmi Narayan Sridhar^*

Published in International Journal of Systems Science and Applied Mathematics (Volume 9, Issue 3)

Received: 1 October 2024 Accepted: 17 October 2024 Published: 11 November 2024

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Abstract

Objective: All optimal control work involving ecological models involves single objective optimization. In this work, we perform multiobjective nonlinear model predictive control (MNLMPC) in conjunction with bifurcation analysis on an ecosystem model. Methods: Bifurcation analysis was performed using the MATLAB software MATCONT while the multi-objective nonlinear model predictive control was performed by using the optimization language PYOMO. Results: Rigorous proof showing the existence of bifurcation (branch) points is presented along with computational validation. It is also demonstrated (both numerically and analytically) that the presence of the branch points was instrumental in obtaining the Utopia solution when the multiobjective nonlinear model prediction calculations were performed. Conclusions: The main conclusions of this work are that one can attain the utopia point in MNLMPC calculations because of the branch points that occur in the ecosystem model and the presence of the branch point can be proved analytically. The use of rigorous mathematics to enhance sustainability will be a significant step in encouraging sustainable development. The main practical implication of this work is that the strategies developed here can be used by all researchers involved in maximizing sustainability The future work will involve using these mathematical strategies to other ecosystem models and food chain models which will be a huge step in developing strategies to address problems involving nutrition.

Published in	International Journal of Systems Science and Applied Mathematics (Volume 9, Issue 3)
DOI	10.11648/j.ijssam.20240903.11
Page(s)	37-43
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Ecosystem, Bifurcation, Optimal Control

1. Introduction

Sustainability is a significant factor to be considered in almost all physical and chemical phenomena. Beneficial activities and situations must be sustained over a considerable amount of time. This is especially true in ecosystem management where the conservation of natural species is essential for ensuring a healthy environment for the long-term well-being of the human population. The issue of sustainability should be implemented in optimization and control studies of ecosystems.

Cabezas and co-workers

[1-9]

have applied the fisher index

[10]

as a sustainability criterion for ecosystems. Specifically, the sustainability concept has been applied in the management of ecosystems, by controlling the population of various species.

Shastri and DIwekar

[11]

and Sorayya et al

[12]

performed single objective optimal control calculations on ecological models maximizing the fisher index to ensure maximum sustainability.

This work aims to perform bifurcation analysis and multiobjective nonlinear model predictive control calculations. The bifurcation analysis reveals the existence of branch points. A rigorous mathematical analysis (which is also computationally validated) demonstrating the existence of branch points is presented. It is shown that the presence of the branch points makes the multiobjective nonlinear model predictive control calculations to reach the utopia solution. This demonstrates that one can maximize the conservation of the natural habitat and maintain maximum sustainability.

2. Equations in Ecological Model

The equations are the following

(1)

(2)

The base parameter values are

3. Computational Procedures Used

Bifurcation analysis

Multiple steady-state solutions are caused by a) Branch Points and b) limit points. At these points the Jacobian matrix of the steady-state equations has a determinant of 0. At a branch point there are 2 distinct tangents while at a limit point, there is only one tangent at the singular point CL_MATCONT

[13,

14] is commonly used to detect branch and limit points. Here a continuation procedure implementing the Moore-Penrose matrix pseudo-inverse is used. CL_MATCONT obtains the branches of the solutions starting from the bifurcation points.

For an ODE system

(3)

Where

Let the tangent plane at any point x be

. Consider a matrix A as

(4)

The matrix A can be written as

(5)

The tangent surface must satisfy the equation

(6)

For limit and branch points the matrix B must be singular. For a limit point (LP) the

n+1 ^th component of the tangent vector

= 0 and for a branch point (BP) the matrix

must be singular

[15-17]

Multiobjective Nonlinear Model Predictive Control Algorithm

The MNLMPC (multiobjective nonlinear model predictive control) strategy

[18, 19]

used in this work does not involve the use of weighting functions or impose additional constraints

[20]

. For a problem that is posed as

(7)

The MNLMPC method first solves dynamic optimization problems independently minimizing/maximizing each any variable

individually. The minimization/maximization of

will lead to the values

. Then the optimization problem that will be solved is

(8)

This will provide the control values for each time value. The first obtained control value is implemented and the remaining are discarded. This procedure is repeated until the implemented and the first obtained control value are the same.

Pyomo

[21]

, was used for the calculations. Here the differential equations are converted to a Nonlinear Program (NLP) using the orthogonal collocation method

[22]

. The Lagrange-Radau quadrature with three collocation points is used and 10 finite elements are used to solve the optimal control problems. The resulting nonlinear optimization problem was solved using the solvers IPOPT

[23]

, and confirmed as global solutions with BARON

[24]

The calculations are repeated until there is no difference between the implemented and the first obtained control values The Utopia point is when

for all i.

Effect of singularities (Limit Point (LP) and Branch Point (BP)) on MNLMPC

If the minimization of the variables

l result in the values

and

the resulting optimization problem will be

(9)

Taking

is the lagrangian multiplier., the Euler Lagrange equation(costate equations) will be

(10)

the derivative of the objective function will yield

(11)

At the Utopia point both

and

are zero. Hence

(12)

The co-state equation in optimal control is

(13)

is the lagrangian multiplier. The first term in this equation is 0 and hence

(14)

yields a limit or a branch point,

is singular.

This implies that there are two different vectors-values for

where

and

. In between there is a vector

where

. This coupled with the boundary condition

will lead to

which will cause the problem to become unconstrained. The only solution for the unconstrained problem is the Utopia solution. This is illustrated numerically in the next few sections.

4. Results and Discussion

Bifurcation Analysis of Ecological Model

In the first case, d3 was the bifurcation parameter while k was the bifurcation parameter

in the second case. Figures 1 and 2 show the bifurcation diagrams.

In both instances, there are branch points from which two different branches originate

The derivatives of

with respect to the variables

are

(15)

The Jacobian matrix is

(16)

The determinant is given by

(17)

For steady-state to be attained

This implies that

and/or

If both these terms are 0 det(J)=0 and the Jacobian matrix is singular. This is the only singular point because

and det (J)=0 t will imply that

and det (J)=0 and

will imply that

This singular point will be a branch point with 2 branches that

and

Computational Validation

Case 1 d₃ bifurcation parameter

At the branch point (singular point)

b3 = 250; the value of

Case 2 K is a bifurcation parameter

At the branch point (singular point)

d₃ = 0.04;

b3 = 250; the value of

In both cases, at the singular point,

and.

Figures 1 and 2 show the bifurcation diagrams when d3 and K are the bifurcation parameters.

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Figure 1. Bifurcation diagram with d3 as bifurcation parameter.

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Figure 2. Bifurcation diagram with K as bifurcation parameter.

MNLMPC of the ecological model

The averaged fisher index (FI) is given by

(18)

The expressions of the functions

and the derivatives

are provided in equation sets 2 and 3. Both d3 and k were used as control variables. Both

and the Fisher index (FI) were maximized individually. The maximization of

resulted in a value of 716.534 while the maximization of FI resulted in a value of 3.965e-05. For the multiobjective nonlinear model predictive calculations, the function minimized was

subject to the equation set 2. The resulting objective function value obtained was the utopia point 0. The multiobjective nonlinear model control variables obtained were d3 = 0.0274 and k 680.00.

Figures 3-6 show the profiles for the MNLMPC calculations.

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Figure 3. X1, X2, X3 profiles for MNLMPC calculations.

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Figure 4. FI versus t.

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Figure 5. d3 versus t.

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Figure 6. K versus t.

5. Conclusions

The main conclusions of this work are that one can attain the utopia point in MNLMPC calculations because of the branch points that occur in the ecosystem model and the prese

nce of the branch point can be proved analytically. The use of rigorous mathematics to enhance sustainability will be a significant step in encouraging sustainable development. The main practical implication of this work is that the strategies developed here can be used by all researchers involved in maximizing sustainability The future work will involve using these mathematical strategies to other ecosystem models and food chain models which will be a huge step in developing strategies to address problems involving nutrition.

Nomenclature

	Prey Population
	Predator Population
	Super Predator Population
r	Prey Growrh Rate
K	Predator Growth rate
	The Maximum Predation rate of Predator and Super Predator
	Half Saturation Constant of Predator and Super Predator
	Death of Predator and Super Predator
FI	Fisher Index
BP	Branch Point
LP	Limit Point
MNLMPC	Multi-objective Nonlinear Model Predictive Control

Author Contributions

Lakshmi Narayan Sridhar is the sole author. The author read and approved the final manuscript.

Data Availability

All data used is presented in the paper.

Conflicts of Interest

The author declares no conflicts of interest.

References

[1]	Ahmad, N., Derrible, S., Eason, T., Cabezas, H., 2016. Using fisher information to track stability in multivariate systems. R. Soc. Open Sci. 3 160582.
[2]	Cabezas, H., Fath, B. D., 2002. Towards a theory of sustainable systems. Fluid Phase Equilib. 194–197, 3–14.
[3]	Cabezas, H., Pawlowski, C. W., Mayer, A. L., Hoagland, N. T., 2003. Sustainability: Ecological, social, economic, technological, and systems perspectives. Clean Tech. Environ. Policy 5, 167–180.
[4]	Cabezas, H., Pawlowski, C. W., Mayer, A. L., Hoagland, N. T., 2005. Simulated experiments with complex sustainable systems: Ecology and technology. Resour. Conserv. Recycl. 44, 279–291.
[5]	Cabezas, H., Pawlowski, C. W., Mayer, A. L., Hoagland, N. T., 2005. Sustainable systems theory: Ecological and other aspects. J. Cleaner Prod. 13, 455–467.
[6]	Cabezas, H., Whitmore, H. W., Pawlowski, C. W., Mayer, A. L., 2007. On the sustainability of an integral model system with industrial, ecological, and macroeconomic components. Resour. Conserv. Recycl. 50, 122–129.
[7]	Doshi, R., Diwekar, U., Benavides, P. T., Yenkie, K. M., Cabezas, H., 2015. Maximizing sustainability of ecosystem model through socio-economic policies derived from multivariable optimal control theory. Clean Techn. Environ. Policy 17, 1573–1583.
[8]	Fath, B. D., Cabezas, H., 2004. Exergy and fisher information as ecological indices. Ecol. Modell. 174, 25–35.
[9]	Fath, B. D., Cabezas, H., Pawlowski, C. W., 2003. Regime changes in ecological systems: An information theory approach. J. Theor. Biol. 222, 517–530.
[10]	Fisher, R. A., 1922. On the mathematical foundations of theoretical statistics. Philos. Trans. R. Soc. A 222, 309–368.
[11]	Shastri, Y., U. Diwekar, Sustainable ecosystem management using optimal control theory: Part 1 (deterministic systems), Journal of Theoretical Biology, Volume 241, Issue 3, 2006, Pages 506-521, ISSN 0022-5193 https://doi.org/10.1016/j.jtbi.2005.12.014
[12]	Soraya Rawlings, E., J. C. Barrera-Martinez, Vicente Rico-Ramirez, Fisher information calculation in a complex ecological model: An optimal control-based approach, Ecological Modelling, Volume 416, 2020, 108845, ISSN 0304-3800, https://doi.org/10.1016/j.ecolmodel.2019.108845
[13]	Dhooge, A., Govearts, W., and Kuznetsov, A. Y., MATCONT: A Matlab package for numerical bifurcation analysis of ODEs, ACM transactions on Mathematical software 29(2) pp. 141-164, 2003.
[14]	Dhooge, A., W. Govaerts; Y. A. Kuznetsov, W. Mestrom, and A. M. Riet, CL_MATCONT; A continuation toolbox in Matlab, 2004.
[15]	Kuznetsov, Y. A. Elements of applied bifurcation theory. Springer, NY, 1998.
[16]	Kuznetsov, Y. A.(2009). Five lectures on numerical bifurcation analysis, Utrecht University, NL., 2009.
[17]	Govaerts, w. J. F., Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, 2000.
[18]	Flores-Tlacuahuac, A. Pilar Morales and Martin Riveral Toledo; Multiobjective Nonlinear model predictive control of a class of chemical reactors. I & EC research; 5891-5899, 2012.
[19]	Sridhar, Lakshmi N., (2019) Multiobjective optimization and nonlinear model predictive control of the continuous fermentation process involving Saccharomyces Cerevisiae, Biofuels, https://doi.org/10.1080/17597269.2019.1674000
[20]	Miettinen, Kaisa, M., Nonlinear Multiobjective Optimization; Kluwers international series, 1999.
[21]	Hart, William E., Carl D. Laird, Jean-Paul Watson, David L. Woodruff, Gabriel A. Hackebeil, Bethany L. Nicholson, and John D. Siirola. Pyomo – Optimization Modeling in Python. Second Edition. Vol. 67. Springer, 2017.
[22]	Biegler, L. T. An overview of simultaneous strategies for dynamic optimization. Chem. Eng. Process. Process Intensif. 46, 1043–105 (2007).
[23]	Wächter, A., Biegler, L. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106, 25–57 (2006). https://doi.org/10.1007/s10107-004-0559-y
[24]	Tawarmalani, M. and N. V. Sahinidis, A polyhedral branch-and-cut approach to global optimization, Mathematical Programming, 103(2), 225-249, 200.

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Sridhar, L. N. (2024). Bifurcation Analysis and Multiobjective Nonlinear Model Predictive Control of Sustainable Ecosystems. International Journal of Systems Science and Applied Mathematics, 9(3), 37-43. https://doi.org/10.11648/j.ijssam.20240903.11

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ACS Style

Sridhar, L. N. Bifurcation Analysis and Multiobjective Nonlinear Model Predictive Control of Sustainable Ecosystems. Int. J. Syst. Sci. Appl. Math. 2024, 9(3), 37-43. doi: 10.11648/j.ijssam.20240903.11

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AMA Style

Sridhar LN. Bifurcation Analysis and Multiobjective Nonlinear Model Predictive Control of Sustainable Ecosystems. Int J Syst Sci Appl Math. 2024;9(3):37-43. doi: 10.11648/j.ijssam.20240903.11

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@article{10.11648/j.ijssam.20240903.11,
  author = {Lakshmi Narayan Sridhar},
  title = {Bifurcation Analysis and Multiobjective Nonlinear Model Predictive Control of Sustainable Ecosystems
},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {9},
  number = {3},
  pages = {37-43},
  doi = {10.11648/j.ijssam.20240903.11},
  url = {https://doi.org/10.11648/j.ijssam.20240903.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20240903.11},
  abstract = {Objective: All optimal control work involving ecological models involves single objective optimization. In this work, we perform multiobjective nonlinear model predictive control (MNLMPC) in conjunction with bifurcation analysis on an ecosystem model. Methods: Bifurcation analysis was performed using the MATLAB software MATCONT while the multi-objective nonlinear model predictive control was performed by using the optimization language PYOMO. Results: Rigorous proof showing the existence of bifurcation (branch) points is presented along with computational validation. It is also demonstrated (both numerically and analytically) that the presence of the branch points was instrumental in obtaining the Utopia solution when the multiobjective nonlinear model prediction calculations were performed. Conclusions: The main conclusions of this work are that one can attain the utopia point in MNLMPC calculations because of the branch points that occur in the ecosystem model and the presence of the branch point can be proved analytically. The use of rigorous mathematics to enhance sustainability will be a significant step in encouraging sustainable development. The main practical implication of this work is that the strategies developed here can be used by all researchers involved in maximizing sustainability The future work will involve using these mathematical strategies to other ecosystem models and food chain models which will be a huge step in developing strategies to address problems involving nutrition.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - Bifurcation Analysis and Multiobjective Nonlinear Model Predictive Control of Sustainable Ecosystems

AU  - Lakshmi Narayan Sridhar
Y1  - 2024/11/11
PY  - 2024
N1  - https://doi.org/10.11648/j.ijssam.20240903.11
DO  - 10.11648/j.ijssam.20240903.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 37
EP  - 43
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20240903.11
AB  - Objective: All optimal control work involving ecological models involves single objective optimization. In this work, we perform multiobjective nonlinear model predictive control (MNLMPC) in conjunction with bifurcation analysis on an ecosystem model. Methods: Bifurcation analysis was performed using the MATLAB software MATCONT while the multi-objective nonlinear model predictive control was performed by using the optimization language PYOMO. Results: Rigorous proof showing the existence of bifurcation (branch) points is presented along with computational validation. It is also demonstrated (both numerically and analytically) that the presence of the branch points was instrumental in obtaining the Utopia solution when the multiobjective nonlinear model prediction calculations were performed. Conclusions: The main conclusions of this work are that one can attain the utopia point in MNLMPC calculations because of the branch points that occur in the ecosystem model and the presence of the branch point can be proved analytically. The use of rigorous mathematics to enhance sustainability will be a significant step in encouraging sustainable development. The main practical implication of this work is that the strategies developed here can be used by all researchers involved in maximizing sustainability The future work will involve using these mathematical strategies to other ecosystem models and food chain models which will be a huge step in developing strategies to address problems involving nutrition.

VL  - 9
IS  - 3
ER  -

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Author Information

Lakshmi Narayan Sridhar

Department of Chemical Engineering, University of Puerto Rico, Mayaguez, Puerto Rico

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Sridhar, L. N. (2024). Bifurcation Analysis and Multiobjective Nonlinear Model Predictive Control of Sustainable Ecosystems. International Journal of Systems Science and Applied Mathematics, 9(3), 37-43. https://doi.org/10.11648/j.ijssam.20240903.11

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ACS Style

Sridhar, L. N. Bifurcation Analysis and Multiobjective Nonlinear Model Predictive Control of Sustainable Ecosystems. Int. J. Syst. Sci. Appl. Math. 2024, 9(3), 37-43. doi: 10.11648/j.ijssam.20240903.11

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AMA Style

Sridhar LN. Bifurcation Analysis and Multiobjective Nonlinear Model Predictive Control of Sustainable Ecosystems. Int J Syst Sci Appl Math. 2024;9(3):37-43. doi: 10.11648/j.ijssam.20240903.11

Copy | Download

@article{10.11648/j.ijssam.20240903.11,
  author = {Lakshmi Narayan Sridhar},
  title = {Bifurcation Analysis and Multiobjective Nonlinear Model Predictive Control of Sustainable Ecosystems
},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {9},
  number = {3},
  pages = {37-43},
  doi = {10.11648/j.ijssam.20240903.11},
  url = {https://doi.org/10.11648/j.ijssam.20240903.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20240903.11},
  abstract = {Objective: All optimal control work involving ecological models involves single objective optimization. In this work, we perform multiobjective nonlinear model predictive control (MNLMPC) in conjunction with bifurcation analysis on an ecosystem model. Methods: Bifurcation analysis was performed using the MATLAB software MATCONT while the multi-objective nonlinear model predictive control was performed by using the optimization language PYOMO. Results: Rigorous proof showing the existence of bifurcation (branch) points is presented along with computational validation. It is also demonstrated (both numerically and analytically) that the presence of the branch points was instrumental in obtaining the Utopia solution when the multiobjective nonlinear model prediction calculations were performed. Conclusions: The main conclusions of this work are that one can attain the utopia point in MNLMPC calculations because of the branch points that occur in the ecosystem model and the presence of the branch point can be proved analytically. The use of rigorous mathematics to enhance sustainability will be a significant step in encouraging sustainable development. The main practical implication of this work is that the strategies developed here can be used by all researchers involved in maximizing sustainability The future work will involve using these mathematical strategies to other ecosystem models and food chain models which will be a huge step in developing strategies to address problems involving nutrition.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - Bifurcation Analysis and Multiobjective Nonlinear Model Predictive Control of Sustainable Ecosystems

AU  - Lakshmi Narayan Sridhar
Y1  - 2024/11/11
PY  - 2024
N1  - https://doi.org/10.11648/j.ijssam.20240903.11
DO  - 10.11648/j.ijssam.20240903.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 37
EP  - 43
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20240903.11
AB  - Objective: All optimal control work involving ecological models involves single objective optimization. In this work, we perform multiobjective nonlinear model predictive control (MNLMPC) in conjunction with bifurcation analysis on an ecosystem model. Methods: Bifurcation analysis was performed using the MATLAB software MATCONT while the multi-objective nonlinear model predictive control was performed by using the optimization language PYOMO. Results: Rigorous proof showing the existence of bifurcation (branch) points is presented along with computational validation. It is also demonstrated (both numerically and analytically) that the presence of the branch points was instrumental in obtaining the Utopia solution when the multiobjective nonlinear model prediction calculations were performed. Conclusions: The main conclusions of this work are that one can attain the utopia point in MNLMPC calculations because of the branch points that occur in the ecosystem model and the presence of the branch point can be proved analytically. The use of rigorous mathematics to enhance sustainability will be a significant step in encouraging sustainable development. The main practical implication of this work is that the strategies developed here can be used by all researchers involved in maximizing sustainability The future work will involve using these mathematical strategies to other ecosystem models and food chain models which will be a huge step in developing strategies to address problems involving nutrition.

VL  - 9
IS  - 3
ER  -

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